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## G = C2×C8⋊D13order 416 = 25·13

### Direct product of C2 and C8⋊D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C8⋊D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — C2×C4×D13 — C2×C8⋊D13
 Lower central C13 — C26 — C2×C8⋊D13
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C2×C8⋊D13
G = < a,b,c,d | a2=b8=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 400 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C13, C2×C8, C2×C8, M4(2), C22×C4, D13, C26, C26, C2×M4(2), Dic13, C52, D26, D26, C2×C26, C132C8, C104, C4×D13, C2×Dic13, C2×C52, C22×D13, C8⋊D13, C2×C132C8, C2×C104, C2×C4×D13, C2×C8⋊D13
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D13, C2×M4(2), D26, C4×D13, C22×D13, C8⋊D13, C2×C4×D13, C2×C8⋊D13

Smallest permutation representation of C2×C8⋊D13
On 208 points
Generators in S208
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)
(1 196 40 170 14 183 27 157)(2 197 41 171 15 184 28 158)(3 198 42 172 16 185 29 159)(4 199 43 173 17 186 30 160)(5 200 44 174 18 187 31 161)(6 201 45 175 19 188 32 162)(7 202 46 176 20 189 33 163)(8 203 47 177 21 190 34 164)(9 204 48 178 22 191 35 165)(10 205 49 179 23 192 36 166)(11 206 50 180 24 193 37 167)(12 207 51 181 25 194 38 168)(13 208 52 182 26 195 39 169)(53 105 92 144 66 118 79 131)(54 106 93 145 67 119 80 132)(55 107 94 146 68 120 81 133)(56 108 95 147 69 121 82 134)(57 109 96 148 70 122 83 135)(58 110 97 149 71 123 84 136)(59 111 98 150 72 124 85 137)(60 112 99 151 73 125 86 138)(61 113 100 152 74 126 87 139)(62 114 101 153 75 127 88 140)(63 115 102 154 76 128 89 141)(64 116 103 155 77 129 90 142)(65 117 104 156 78 130 91 143)
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143)(144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169)(170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(183 208)(184 207)(185 206)(186 205)(187 204)(188 203)(189 202)(190 201)(191 200)(192 199)(193 198)(194 197)(195 196)

G:=sub<Sym(208)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208), (1,196,40,170,14,183,27,157)(2,197,41,171,15,184,28,158)(3,198,42,172,16,185,29,159)(4,199,43,173,17,186,30,160)(5,200,44,174,18,187,31,161)(6,201,45,175,19,188,32,162)(7,202,46,176,20,189,33,163)(8,203,47,177,21,190,34,164)(9,204,48,178,22,191,35,165)(10,205,49,179,23,192,36,166)(11,206,50,180,24,193,37,167)(12,207,51,181,25,194,38,168)(13,208,52,182,26,195,39,169)(53,105,92,144,66,118,79,131)(54,106,93,145,67,119,80,132)(55,107,94,146,68,120,81,133)(56,108,95,147,69,121,82,134)(57,109,96,148,70,122,83,135)(58,110,97,149,71,123,84,136)(59,111,98,150,72,124,85,137)(60,112,99,151,73,125,86,138)(61,113,100,152,74,126,87,139)(62,114,101,153,75,127,88,140)(63,115,102,154,76,128,89,141)(64,116,103,155,77,129,90,142)(65,117,104,156,78,130,91,143), (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)>;

G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208), (1,196,40,170,14,183,27,157)(2,197,41,171,15,184,28,158)(3,198,42,172,16,185,29,159)(4,199,43,173,17,186,30,160)(5,200,44,174,18,187,31,161)(6,201,45,175,19,188,32,162)(7,202,46,176,20,189,33,163)(8,203,47,177,21,190,34,164)(9,204,48,178,22,191,35,165)(10,205,49,179,23,192,36,166)(11,206,50,180,24,193,37,167)(12,207,51,181,25,194,38,168)(13,208,52,182,26,195,39,169)(53,105,92,144,66,118,79,131)(54,106,93,145,67,119,80,132)(55,107,94,146,68,120,81,133)(56,108,95,147,69,121,82,134)(57,109,96,148,70,122,83,135)(58,110,97,149,71,123,84,136)(59,111,98,150,72,124,85,137)(60,112,99,151,73,125,86,138)(61,113,100,152,74,126,87,139)(62,114,101,153,75,127,88,140)(63,115,102,154,76,128,89,141)(64,116,103,155,77,129,90,142)(65,117,104,156,78,130,91,143), (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169)(170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196) );

G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,141),(38,142),(39,143),(40,144),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,161),(58,162),(59,163),(60,164),(61,165),(62,166),(63,167),(64,168),(65,169),(66,170),(67,171),(68,172),(69,173),(70,174),(71,175),(72,176),(73,177),(74,178),(75,179),(76,180),(77,181),(78,182),(79,183),(80,184),(81,185),(82,186),(83,187),(84,188),(85,189),(86,190),(87,191),(88,192),(89,193),(90,194),(91,195),(92,196),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208)], [(1,196,40,170,14,183,27,157),(2,197,41,171,15,184,28,158),(3,198,42,172,16,185,29,159),(4,199,43,173,17,186,30,160),(5,200,44,174,18,187,31,161),(6,201,45,175,19,188,32,162),(7,202,46,176,20,189,33,163),(8,203,47,177,21,190,34,164),(9,204,48,178,22,191,35,165),(10,205,49,179,23,192,36,166),(11,206,50,180,24,193,37,167),(12,207,51,181,25,194,38,168),(13,208,52,182,26,195,39,169),(53,105,92,144,66,118,79,131),(54,106,93,145,67,119,80,132),(55,107,94,146,68,120,81,133),(56,108,95,147,69,121,82,134),(57,109,96,148,70,122,83,135),(58,110,97,149,71,123,84,136),(59,111,98,150,72,124,85,137),(60,112,99,151,73,125,86,138),(61,113,100,152,74,126,87,139),(62,114,101,153,75,127,88,140),(63,115,102,154,76,128,89,141),(64,116,103,155,77,129,90,142),(65,117,104,156,78,130,91,143)], [(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143),(144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169),(170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(157,182),(158,181),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(183,208),(184,207),(185,206),(186,205),(187,204),(188,203),(189,202),(190,201),(191,200),(192,199),(193,198),(194,197),(195,196)]])

116 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 13A ··· 13F 26A ··· 26R 52A ··· 52X 104A ··· 104AV order 1 2 2 2 2 2 4 4 4 4 4 4 8 8 8 8 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 26 26 1 1 1 1 26 26 2 2 2 2 26 26 26 26 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 M4(2) D13 D26 D26 C4×D13 C4×D13 C8⋊D13 kernel C2×C8⋊D13 C8⋊D13 C2×C13⋊2C8 C2×C104 C2×C4×D13 C4×D13 C2×Dic13 C22×D13 C26 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 4 6 12 6 12 12 48

Matrix representation of C2×C8⋊D13 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 288 0 0 0 0 288 0 0 0 0 294 161 0 0 262 19
,
 85 312 0 0 1 0 0 0 0 0 13 312 0 0 220 248
,
 85 312 0 0 25 228 0 0 0 0 112 37 0 0 287 201
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[288,0,0,0,0,288,0,0,0,0,294,262,0,0,161,19],[85,1,0,0,312,0,0,0,0,0,13,220,0,0,312,248],[85,25,0,0,312,228,0,0,0,0,112,287,0,0,37,201] >;

C2×C8⋊D13 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_{13}
% in TeX

G:=Group("C2xC8:D13");
// GroupNames label

G:=SmallGroup(416,121);
// by ID

G=gap.SmallGroup(416,121);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,50,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽